Inverse Functions Graph Square/Cube Root Functions Objectives: 1.To find the inverse of a function 2.To graph inverse functions 3.To graph square and cube root functions as transformations on parent functions

You will be able to find the inverse of a function

Function Composition Function composition Function composition happens when we take a whole function and substitute it in for x in another function. –The “interior” function gets substituted in for x in the “exterior” function

Exercise 1 Let f ( x ) = 4 x + 2 and g ( x ) = 1/4 x – 1/2. Find the following compositions. 1. f ( g ( x ))2. g ( f ( x ))

Inverse Relations inverse relation An inverse relation is a relation that switches the inputs and output of another relation. Inverse relations “undo” each other

Inverse Functions inverse functions If a relation and its inverse are both functions, then they are called inverse functions. f -1 = “ f inverse” or “inverse of f ”

The inverse of a function is not necessarily a function.

Inverse Functions inverse functions If a relation and its inverse are both functions, then they are called inverse functions.

Exercise 2 Verify that f ( x ) = 2 x + 3 and f -1 ( x ) = ½ x – 3/2 are inverse functions.

Step 1 Step 2 Step 1 Finding the Inverse of a Function

Exercise 3 Find the inverse of f ( x ) = − (2/3) x + 2.

Exercise 4

Exercise 5 Find the inverse of the given function.

Inverses of Nonlinear Functions For some nonlinear functions, you have to first restrict the domain to find its inverse. ?

Inverses of Nonlinear Functions For some nonlinear functions, you have to first restrict the domain to find its inverse.

Inverses of Nonlinear Functions For some nonlinear functions, you have to first restrict the domain to find its inverse.

Exercise 6 Find the inverse of the given function.

You will be able to graph the inverse functions

Graphing Investigation Suppose we drew a triangle on the coordinate plane. Geometrically speaking, what would happen if we switched the x - and the y -coordinates?

Graphing Investigation Suppose we drew a triangle on the coordinate plane. Geometrically speaking, what would happen if we switched the x - and the y -coordinates? This is what happens with inverses.

Graphs of Inverse Functions

Exercise 7

Exercise 8 Let f ( x ) = ½ x – 5. 1.Find f -1 2.State the domain of each function 3.Graph f and f -1 on the same coordinate plane

Exercise 9a 1.For an input of 2, what is the output? Is it unique? 2.For an output of 8, what was the input? Is it unique? 3.What does the answer to Q2 tell you about the inverse of the function?

Exercise 9b Let f ( x ) = x 3. 1.Find f -1 2.State the domain of each function 3.Graph f and f -1 on the same coordinate plane

Exercise 10a 1.For an input of 4, what is the output? Is it unique? 2.For an output of 16, what was the input? Is it unique? 3.What does the answer to Q2 tell you about the inverse of the function?

Exercise 10b Let f ( x ) = x 2. 1.Find f -1 2.State the domain of each function 3.Graph f and f -1 on the same coordinate plane

Does it Function? As the previous Exercise demonstrated, even though you can find the inverse of a function, the inverse itself may not be a function. Remember, we overcome this shortcoming by restricting the domain of the original function.

Inverses of Nonlinear Functions For some nonlinear functions, you have to first restrict the domain to find its inverse. ?

Inverses of Nonlinear Functions For some nonlinear functions, you have to first restrict the domain to find its inverse.

Inverses of Nonlinear Functions For some nonlinear functions, you have to first restrict the domain to find its inverse.

Does it Function? Recall that we can use the vertical line test to see if a graph represents a function. Function Not a Function The question is: How can we tell if a function’s inverse will be a function?

Horizontal Line Test The inverse of a function f is also a function iff no horizontal line intersects the graph of f more than once.

Exercise 11 Graph the function f. Then use the graph to determine whether f -1 is a function.

One-to-One Function one-to-one function If f passes both the vertical and the horizontal line tests— that is, if f and f -1 are functions—then f is a one-to-one function. Every input has exactly one output Every output has exactly one input

Radical Parents We can perform transformations on these parent functions to help graph whole families of radical functions.

Radical Parents Use the following to discover the roll of a, h, and k in the following radical functions: a Scaling 0 < | a | < 1: Shrink vertically | a | > 1: Stretch vertically a < 0: Flips h Horizontal translation k Vertical translation

Exercise 12 Graph the following radical function. Then state the domain and range.

Exercise 13 Graph the following radical function. Then state the domain and range.

Exercise 14 Graph the following radical functions. Then state the domain and range. 1.

Inverse Functions Graph Square/Cube Root Functions Objectives: 1.To find the inverse of a function 2.To graph inverse functions 3.To graph square and cube root functions as transformations on parent functions